\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 113, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2008/113\hfil Higher order neutral differential equations]
{Oscillation and asymptotic behaviour of a higher order neutral differential
equation with positive and negative coefficients}
\author[B. Karpuz, L. N. Padhy, R. N. Rath \hfil EJDE-2008/113\hfilneg]
{Ba\c{s}ak Karpuz, Laxmi Narayan Padhy, Radhanath Rath}
\address{Ba\c{s}ak Karpuz \newline
Department of Mathematics, Facaulty of Science and arts, A.N.S.
Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey}
\email{bkarpuz@gmail.com}
\address{Laxmi Narayan Padhy \newline
Department Of Computer Science and Engineering,
K.I.S.T., Bhubaneswar, Orissa, India}
\email{ln\_padhy\_2006@yahoo.co.in}
\address{Radhanath Rath \newline
Department of Mathematics, Khallikote Autonomous College,
Berhampur, 760001 Orissa, India}
\email{radhanathmath@yahoo.co.in}
\thanks{Submitted April 4, 2008. Published August 20, 2008.}
\subjclass[2000]{34C10, 34C15, 34K40}
\keywords{Oscillatory solution; neutral differential equation;
\hfill\break\indent asymptotic behaviour}
\begin{abstract}
In this paper, we obtain necessary and sufficient conditions so
that every solution of
$$
\big(y(t)- p(t) y(r(t))\big)^{(n)}+
q(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t)
$$
oscillates or tends to zero as $t \to \infty$, where $n$ is
an integer $n \geq 2$, $q>0$, $u\geq 0$.
Both bounded and unbounded solutions are considered in this paper.
The results hold also when $u\equiv 0$, $f(t)\equiv 0$, and $G(u)\equiv u$.
This paper extends and generalizes some recent results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\section{Introduction}
In this article, we obtain necessary and sufficient conditions
for every solution of the higher-order
neutral functional differential equation
\begin{equation}\label{n50}
\big(y(t)- p(t) y(r(t))\big)^{(n)}+
q(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t)
\end{equation}
to oscillate or to tend to zero as $t$ tends to infinity,
where, $n$ is an integer $n \geq 2$,
$p,f\in C([0,\infty),\mathbb{R})$, $q,u\in C([0,\infty),[0,\infty))$, and
$G, H \in C(\mathbb{R},\mathbb{R})$. The functional delays $r(t), g(t)$ and $h(t)$
are continuous, strictly increasing and unbounded functions for
$t\geq t_0$ such that $r(t) \leq t,g(t)\leq t$, and $h(t)\leq t$.
Some of the following assumptions will be used in this article.
\begin{itemize}
\item[(H0)] $G$ is non-decreasing, $xG(x)>0$ for $x \neq 0$.
\item[(H1)] $\liminf_{t\to\infty}r(t)/t >0$.
\item[(H2)] $H$ is bounded.
\item[(H3)] $\liminf_{|u|\to \infty}G(u)/u\geq \delta$
where $\delta>0$.
\item[(H4)] $ \int_{t_0}^{\infty}t^{n-2}q(t)\,dt = \infty$, $n \geq 2$.
\item[(H5)] $ \int_{t_0}^{\infty}t^{n-1}u(t)\,dt < \infty$.
\item[(H6)] $ \int_{t_0}^{\infty}t^{n-1}q(t)\,dt = \infty$.
\item[(H7)] There exists a bounded function $F\in C^{(n)}([0,\infty),\mathbb{R})$ such
that $F^{(n)}(t)=f(t)$ and $\lim_{t\to \infty}F(t)=0$.
\item[(H8)]There exists a bounded function $F\in C^{(n)}([0,\infty),\mathbb{R})$ such
that $F^{(n)}(t)=f(t)$.
\item[(H9)] $\liminf_{t\to\infty}g(t)/t >0$.
\item[(H10)] $\int^\infty q(t)\,dt=\infty$
\end{itemize}
Note that, we do not need the condition
``$xH(x)>0$ for $x \neq 0$" in the proofs of our results.
However, one may assume them for technical reasons; i.e,
to make \eqref{n50} a neutral equation with positive and negative
coefficients. Further, we note that, for $n \geq 2$, condition
(H10) implies (H4) and furthermore, (H4) implies (H6).
For $\tau ,\sigma ,\alpha$ positive constants,
we put $r(t)=t-\tau$,
$g(t)=t-\sigma$ and $h(t)=t-\alpha$. Then \eqref{n50} reduces to
\begin{equation} \label{e50}
\big(y(t)- p(t) y(t-\tau)\big)^{(n)}+
q(t)G( y(t-\sigma))-u(t)H(y(t-\alpha)) = f(t)\,.
\end{equation}
If $n=1$ then \eqref{e50} reduces to
\begin{equation} \label{ene56}
\big(y(t)- p(t) y(t-\tau)\big)'+
q(t) G(y(t-\sigma))-u(t)G(y(t-\alpha)) =f(t),
\end{equation}
which was studied in \cite{rat}.
Our objective is to generalize the results in
\cite{rat} to the higher-order equation \eqref{n50}.
Further, if $u=0$, then \eqref{n50} takes the form
\begin{equation}\label{ene49}
\big(y(t)- p(t) y(r(t))\big)^{(n)}+
q(t)G( y(g(t))) = f(t).
\end{equation}
Hence \eqref{e50}--\eqref{ene49} are particular cases of \eqref{n50}.
The authors in \cite[p. 195]{sahi}, suggested the study of
unbounded solutions for \eqref{ene49}, particularly when
$0\leq p(t)\leq p<1$; this paper accomplishes that task.
The motivation of this work came from the fact that almost no work
is done on oscillatory behaviour of unbounded solutions of neutral
differential equations \eqref{e50} of order $n > 2$.
For the case when $n$ is odd, the authors in \cite{Li52}, have presented
a result for the linear equation
\begin{equation} \label{ene50}
\big(y(t)- p(t) y(t-\tau)\big)^{(n)}+
q(t) y(t-\sigma)-u(t)y(t-\alpha) = 0,
\end{equation}
with the assumptions
\begin{itemize}
\item[(AD1)] $q(t)> u(t-\sigma+\alpha)$ and
\item[(AD2)] $\sigma >\alpha$ or $\alpha >\sigma$.
\end{itemize}
In \cite{ocal}, the author obtained sufficient conditions for the
oscillation of solutions of the linear homogeneous equation
\begin{equation} \label{ene55}
\big(y(t)- p(t) y(t-\tau)\big)'+
q(t) y(t-\sigma)-u(t)y(t-\alpha) = 0,
\end{equation}
with the assumptions (AD1) and (AD2).
In \cite{misra10} the authors obtained sufficient conditions for oscillation
of the equation \eqref{ene56}
and other results under the conditions (AD1), (AD2), and
\begin{itemize}
\item[(AD3)] $\liminf_{|u|\rightarrow\infty} G(u)/u 0$.
\end{itemize}
In \cite{manoj} the authors studied a second order neutral equation
with several delay terms of the form
\begin{equation} \label{en53}
\big(y(t)- p(t) y(t-\tau)\big)''+\sum_{i=1}^k
q_i(t) y(t-\sigma_i)-\sum_{i=1}^m u_i(t)y(t-\alpha_i) = f(t).
\end{equation}
When $k=m=1$, the above equation takes the form
\begin{equation} \label{en54}
\big(y(t)- p(t) y(t-\tau)\big)''+
q(t) y(t-\sigma)- u(t)y(t-\alpha) = f(t).
\end{equation}
Here also, the authors require the conditions (AD1), (AD2),
\begin{itemize}
\item[(AD4)] $u(t)0$\,.
\end{itemize}
In this paper, an attempt is made to relax the conditions (AD1)--(AD5)
and study the oscillation and non-oscillation of \eqref{n50}.
Our results hold also when $u\equiv 0, f(t)\equiv 0$, and $G(u) \equiv u$.
As a consequence, this paper extends and generalizes some
of the recent results in \cite{manoj,misra10,rat,sahi}.
Appropriate examples are included to illustrate our results.
Let $t_{0}\geq0$ and $t_{-1}:=\min\{r(t_{0}),g(t_{0}),h(t_{0})\}$.
By a \emph{solution} of \eqref{n50}, we mean a function $y\in C([t_{-1},\infty),\mathbb{R})$ such that $y(t)-p(t)y(r(t))$ is $n$ times continuously differentiable on $[t_{0},\infty)$ and the neutral equation \eqref{n50} is satisfied by $y(t)$ for all $t\geq t_{0}$.
It is known that \eqref{n50} has a unique solution provided that an initial function $\phi\in C([t_{-1},t_{0}],\mathbb{R})$ is given to satisfy $y(t)=\phi(t)$ for all $t\in[t_{-1},t_{0}]$.
Such a solution is said to be \emph{non-oscillatory} if it is eventually positive or eventually negative for large $t$, otherwise it is called \emph{oscillatory}.
In this work we assume the existence of
solutions and study only their qualitative behaviour.
For existence and uniqueness of solutions, the
reader is referred to \cite{gori5}.
In the sequel, unless otherwise specified, when we write a functional
inequality, it will be assumed to hold for all sufficiently large
values of $t$.
\section{Main results}
We assume that $p(t)$ satisfies one of
the following conditions in this work.
\begin{itemize}
\item[(A1)] $0 \leq p(t)\leq p < 1$,
\item[(A2)] $-1 < -p\leq p(t) \leq 0$,
\item[(A3)] $0\leq p(t) \leq p_1$,
\item[(A4)] $-p_2 \leq p(t) \leq -p_1 0 ,\quad \text{for } j=0,1,2,\dots.,m,\; t\geq t_0,\\
(-1)^{n+j-1}y(t)y^{(j)}(t)>0 \quad \text{for }
j=m+1,m+2,\dots, n-1,\; t \geq t_0.
\end{gather*}
\end{lemma}
\begin{lemma}\cite[Lemma 1]{sahi} \label{lem2.2}
Let $u,v,p : [0,\infty)\to \mathbb{R} $ be such that
$u(t) = v(t) - p(t)v(r(t))$, $t\geq T_0$,
where $r(t)$ is a continuous, monotonic increasing and unbounded
function such that $r(t) \leq t$.
Suppose that $p(t)$ satisfies one of the conditions {\rm (A2), (A3), (A4)}.
If $v(t)>0$ for $t\geq 0$ and
$\liminf_{t\to \infty}v(t)=0$ and $\lim_{t\to \infty}u(t)= L$
exists, then $L=0$.
\end{lemma}
\begin{lemma}\label{imp1}\cite[Lema 2.1]{6r}
If $\int_0^\infty t^{n-1}|f(t)|\,dt 0$, eventually.
Then there exists $t_0>T_1$ such that
$y(t)>0$, $y(r(t))>0$, $y(g(t))$ and $y(h(t))>0$ for
$t\geq t_0$.
For simplicity of notation, define for $t\geq t_0$ ,
\begin{equation}\label{e51}
z(t) = y(t) - p(t) y(r(t))\,.
\end{equation}
Further, due to the assumption (H2) and (H5), we define for $t \geq t_0$
\begin{equation}\label{e52}
k(t)=\frac{(-1)^{n-1}}{(n-1)!}\int_{t}^{\infty}(s-t)^{n-1}u(s)H(y(h(s)))\,ds.
\end{equation}
Then
\begin{equation}\label{e613}
k^{(n)}(t)=-u(t)H(y(h(t))).
\end{equation}
Set
\begin{equation}\label{e53}
w(t)=z(t)+k(t)-F(t).
\end{equation}
Then using \eqref{e51}--\eqref{e53} in \eqref{n50}, we obtain
\begin{equation} \label{e54}
w^{(n)}(t) = - q(t)G(y(g(t)))\leq 0.
\end{equation}
Hence $w,w',\dots w^{(n-1)}$ are monotonic and single sign
for $t\geq t_1\geq t_0$.
Then $ \lim_ { t\to\infty}w(t) = \lambda$, where
$-\infty \leq \lambda \leq +\infty$.
From \eqref{e52}, it follows, due to (H2) and (H5) that
\begin{equation}\label{e600}
k(t)\to 0 \quad \text{ as} \quad t\to\infty.
\end{equation}
Since $y(t)$ is unbounded, there exists a sequence $\{a_n\}$ such that
\begin{equation*}
a_n\to\infty, \quad y(a_n)\to\infty, \quad \text{as } n\to\infty,
\end{equation*}
and
\begin{equation}\label{e55}
y(a_n)=\max\{y(s):t_1\leq s \leq a_n\}.
\end{equation}
We may choose $n$ large enough so that for $n \geq N_0$,
$\min\{r(a_n),g(a_n),h(a_n)\}>t_1$.
Then from \eqref{e600} and (H8), it follows that, for $00,w'>0$ for $t\geq t_2 \geq t_1,$.
Since $w^{(n)}(t)\not \equiv 0$ and is non positive, it follows
from Lemma \ref{lemm2.2} that there exists a positive integer $m$
such that $n-m$ is odd and for $t\geq t_3 \geq t_2$, we have
$w^{(j)}(t)>0$ for $j=0,1,\dots,m$ and $w^{(j)}(t)w^{(j+1)}(t)<0$
for $j=m,m+1,\dots,n-2$. Then $\lim_{t\to\infty}w^{(m)}(t)=l$ exists
(as a finite number).
Hence $m\geq 1$. Integrating \eqref{e54}, $n-m$ times from $t$ to $\infty$,
we obtain for $t \geq t_3$
\begin{equation}\label{e57}
w^{(m)}(t)=l+\frac{(-1)^{n-m-1}}{(n-m-1)!}\int_t^\infty(s-t)^{n-m-1}
q(s)G(y(g(s)))\,ds.
\end{equation}
This implies
\begin{equation}\label{e58}
\int_t^\infty(s-t)^{n-m-1}q(s)G(y(g(s)))\,ds0$ such that $g(t)/t \geq b>0$ for large $t$ and since $\lim_{t\to\infty}g(t)=\infty$ then we have
$$
\liminf_{t\to\infty} \frac{y(t)}{t^{m-1}}=0.
$$
Since $m\geq 1$,we can choose $M_0>0$ such that $w(t)>M_0t^{m-1}$ for
$t\geq t_4\geq t_3$.
Thus
\begin{equation}\label{e59}
\liminf_{t\to\infty}\frac{y(t)}{w(t)}=0.
\end{equation}
Set, for $t\geq t_4$,
$$
p^*(t)=p(t)\frac{w(r(t))}{w(t)}.
$$
From (H8), \eqref{e600} and $\lim_{t\to\infty}w(t)=\infty$, we obtain
$$
\lim_{t\to\infty}\frac{(F(t)-k(t))}{w(t)}=0.
$$
Then we have
\begin{equation}\label{e612}
\begin{aligned}
1&=\lim_{t\to\infty}\big[\frac{w(t)}{w(t)}\big] \\
&=\lim_{t\to\infty}\big[\frac{y(t)-p(t)y(r(t))-(F(t)-k(t))}{w(t)}\big] \\
&=\lim_{t\to\infty}\big[\frac{y(t)}{w(t)}-\frac{p^*(t)y(r(t))}{w(r(t))}
-\frac{(F(t)-k(t))}{w(t)}\big]\\
&= \lim_{t\to\infty}\big[\frac{y(t)}{w(t)}-\frac{p^*(t)y(r(t))}{w(r(t))}\big].
\end{aligned}
\end{equation}
Since $w(t)$ is a increasing function, $w(r(t))/w(t)<1$.
If $p(t)$ satisfies (A1) then $0\leq p^*(t)

-1$.
Hence it is clear that if $p(t)$ satisfies (A1) or (A2) then $p^*(t)$
also lies in the ranges (A1) or (A2) accordingly.
Hence use of Lemma \ref{lem2.2} yields, due to \eqref{e59}, that
$$
\lim_{t\to\infty}\big[\frac{y(t)}{w(t)}-\frac{p^*(t)y(r(t))}{w(r(t))}\big]=0,
$$
a contradiction to \eqref{e612}.
Hence the unbounded solution $y(t)$ cannot be eventually positive.
Next, if $y(t)$ is an eventually negative solution of \eqref{n50}
for large $t$, then we set $x(t)=-y(t) $ to obtain $x(t)>0$ and
then \eqref{n50} reduces to
\begin{equation}\label{en1p}
\big(x(t)-p(t)x(\tau(t))\big)+q(t) \tilde{G}(x(g(t)))-u(t)\tilde{H}(x(h(t)))
=\tilde{f}(t)
\end{equation}
where
\begin{equation}\label{en13p}
\tilde{f}(t)=-f(t),\quad \tilde{G}(v)=-G(-v), \quad \tilde{H}(v)=-H(-v).
\end{equation}
Further,
$\tilde{F}(t)=-F(t)$ implies $\tilde{F}^n(t)=\tilde{f}(t)$.
In view of the above facts, it can be easily verified that
the following conditions hold:
\begin{itemize}
\item[(H0')]
$\tilde{G}$ is non-decreasing and
$x\tilde{G}(x)>0$ for $x\neq 0$,
\item[(H2')]
$\tilde{H}$ is bounded,
\item[(H3')]
$\liminf_{|v|\to\infty} \tilde{G}(v)/v\geq \delta >0$,
\item[(H8')]
There exists a bounded function $\tilde{F}(t)$
such that $\tilde{F}^n(t)=\tilde{f}(t)$.
\end{itemize}
Proceeding as in the proof for the case $y(t)>0$, we obtain a contradiction.
Hence $y(t)$ is oscillatory and the proof is complete.
\end{proof}
\begin{remark}\label{rep}\rm
The above theorem answers the open problem
in \cite[p. 195]{sahi}; i.e, to study oscillatory behaviour of unbounded
solutions of \eqref{n50}, when $p(t)$ satisfies (A1).
\end{remark}
The following example illustrates Theorem \ref{thm1}.
\begin{example}\rm
Consider the neutral equation
\begin{align*}
&(y(t)-\alpha y(t-2\pi))''+2e^{-3\pi/2}(e^{2\pi}-\alpha)y(t-\pi/2)
-e^{-4\pi}t^{-2}H(y(t-4\pi))\\
&=\frac{-e^t \cos(t)}{t^2(e^{8\pi}+e^{2t}\cos^2(t))},
\end{align*}
where $0\leq\alpha<1$ or $-1\max\{3,1/(1-\sqrt{p})\}$, where $0